C172 single pilot reported an NMAC event during landing pattern training. Pilot immediately executed an evasive maneuver to avoid a collision.

Narrative

I'm a student pilot training at ZZZ. This incident occurred on a clear day around XA24UTC on Day 0 while practicing touch and goes in the pattern in a C172P. After my second touch and go I was on the runway applying full power for another take-off. While building speed to takeoff before I started rotating I heard the Controller clear an airplane that was holding short to take-off on the same runway I was on. I'd never seen a Controller do this and it seemed unusual at the time but I continued my takeoff. My climbout was normal and I turned crosswind at about 600 AGL. Right before beginning the turn to downwind I heard my traffic alert go off telling me I had close traffic at 8 o'clock. I began my turn to downwind while looking for the traffic expecting it to be in the upwind. Instead the other pilot had made their crosswind turn closer to the runway and had cut me off while climbing. Luckily they were in a SR22 and at a higher altitude than mine but they were right in front of me taking up almost my entire windshield. While they may have been slightly too high for us to collide I feared for my safety considering how close they were and pulled power to idle immediately and initiated a descent. After they were clear of me and continued their crosswind departure I regained my altitude and continued in the pattern as usual. Since I've never had an incident like this and was surprised I did not think to notify the Tower while in the air. Once on the ground; I wanted to discuss it with an experienced pilot; my instructor and the head of flight school. We went over the voice recording and I was educated about this reporting service. My instructor and I located the other pilot and had a debrief call of the incident. They acknowledged the incident as well and confirmed they started their turn at 400 AGL. They shared that they did not see me after I took off. They weren't sure if I was still on the runway when they were cleared but agreed that they could have been more vigilant and less complacent when the Tower cleared them for left turnout and knowing another plane was in the pattern. They also confirmed their traffic alert triggered as well (theirs was older and didn't call my location).We discussed the implications of the classic high-wing / low-wing combination and how it further contributed to this very dangerous situation (they were right wing up while I was on their right with left wing down). Since they were heading to ZZZ1 via the coast; the nearby naval airspace added another element of limited maneuvering space for them as well. On my end; I figured out it would've been good for me to call out that I was in the upwind and turning crosswind and downwind in the same direction they cleared to turn out. Chances are the other more experienced pilot would have reacted to my callout with more vigilance. Also; since I felt the Tower cleared them to takeoff too early I could have also considered aborting my takeoff and questioning it on the ground. Finally; we asked the flight school to share this with the Tower Supervisor so they can learn and improve as well.